Algebraic topology – Calculation of the degree of a map

Let $ q: S ^ n rightarrow S ^ n $ to be the map that quotients the lower hemisphere in the south pole$ s $ . I am asked to calculate the degree of this map.

Yes $ D_ + $ refers to the upper hemisphere and then $ q: D_ + rightarrow S ^ n – s $ is a homeomorphism and so by the formula of the local degree, the degree is $ 1 $ or -1 $.

I've tried to show that this card does not send any points to its antipode and that it is homotopic to the identity. So, he must have a degree $ 1 $.

It's intuitively quite clear to me but I do not know how to write a formal proof.
Any idea / correction / help will be greatly appreciated.